About some properties of similarly homogeneous $\mathbb {R}$-trees

In this paper we consider the properties of locally complete similarly homogeneous inhomogeneous $\mathbb{R}$-trees. The geodesic space is called $\mathbb{R}$-tree if any two points may be connected by the unique arc. The general problem of A. D. Alexandrov on the characterization of metric spaces is considered. The distance one preserving mappings are constructed for some classes of $\mathbb{R}$-trees. To do this, we use the construction with the help of which a new special metric is introduced on an arbitrary metric space. In terms of this new metric, a criterion is formulated that is necessary for a so that a distance one preserving mapping to be isometric. In this case, the characterization by A. D. Alexandrov is not fulfilled. Moreover, the boundary of a strictly vertical $\mathbb{R}$-tree is also studied. It is proved that any horosphere in a strictly vertical $\mathbb{R}$-tree is an ultrametric space. If the branch number of a strictly vertical $\mathbb{R}$-tree is not greater than the continuum, then the cardinality of any sphere and any horosphere in the $\mathbb{R}$-tree equals the continuum, and if the branch number of $\mathbb{R}$-tree is larger than the continuum, then the cardinality of any sphere or horosphere equals the number of branches.